\documentclass[reqno]{amsart} \usepackage{amssymb,hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 143, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/143\hfil Existence of positive solutions] {Existence of positive solutions for \\ nonlinear boundary-value problems in \\ unbounded domains of $\mathbb{R}^{n}$} \author[F. Toumi, N. Zeddini\hfil EJDE-2005/143\hfilneg] {Faten Toumi, Noureddine Zeddini} % in alphabetical order \address{D\'{e}partement de Math\'{e}matiques, Facult\'{e} des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia} \email{Faten.Toumi@fsb.rnu.tn} \email{noureddine.zeddini@ipein.rnu.tn} \date{} \thanks{Submitted September 30, 2005. Published December 8, 2005.} \subjclass[2000]{34B15, 34B27} \keywords{Green function; nonlinear elliptic equation; positive solution; \hfill\break\indent Schauder fixed point theorem} \begin{abstract} Let $D$ be an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$) with a nonempty compact boundary $\partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, \begin{gather*} \Delta u=f(.,u),\quad u>0\quad \text{in }D,\\ u\big|_{\partial D}=\alpha \varphi ,\\ \lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda , \end{gather*} where $\alpha ,\beta,\lambda$ are nonnegative constants with $\alpha +\beta >0$ and $\varphi$ is a nontrivial nonnegative continuous function on $\partial D$. The function $f$ is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and $h$ is a fixed harmonic function in $D$, continuous on $\overline{D}$. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} In this paper, we are concerned with the existence and asymptotic behavior of positive solutions for the following nonlinear elliptic equation, in the sense of distributions, $$\Delta u=f(.,u)\quad\mbox{in } D, \label{E}$$ where $D$ is an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$) with a nonempty compact boundary $\partial D$ and $f$ is a nonnegative measurable function on $D$ that may be singular or sublinear with respect to the second variable. More precisely we will study the problem $$\label{Pab} \begin{gathered} \Delta u=f(.,u),\quad u>0\quad \text{in }D, \\ u\big|_{\partial D}=\alpha \varphi , \\ \lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda \geq 0, \end{gathered}$$ where $\alpha \geq 0$, $\beta \geq 0$, $\varphi$ is a nontrivial nonnegative continuous function on $\partial D$, $h$ is the harmonic function in $D$ given by \eqref{e2.2} below and $f$ satisfies some appropriate conditions related to a Kato class (see Definition \ref{def1}) introduced by Bachar et al in \cite{b1} for $n\geq 3$ and M\^{a}agli and Ma\^{a}toug in \cite{m3} for $n=2$. In \cite{a2}, Athreya considered \eqref{E} with a special case of nonlinearity $f(x,u)=g(u)\leq \max (1,u^{-\alpha })$ for $0<\alpha <1$, on a simply connected bounded $C^{2}$-domain $\Omega$. He showed that if $h_{0}$ is a fixed positive harmonic function in $\Omega$ and $\varphi$ is a nontrivial nonnegative continuous function on $\partial \Omega$, there exists a constant $c>1$ such that if $\varphi \geq c\,h_{0}$ on $\partial \Omega$, then \eqref{E} has a positive continuous solution $u$ satisfying $u=\varphi$ on $\partial \Omega$ and $u\geq h_{0}$ in $\Omega$. This result was extended by Bachar et al \cite{b3} on the half space $\mathbb{R}_{+}^{n}=\{x=(x_{1},\dots ,x_{n})\in \mathbb{R}^{n}:x_{n}>0\}$ ($n\geq 2$). More precisely, they proved that the problem \begin{gather*} \Delta u=f(.,u)\quad \text{in }\mathbb{R}_{+}^{n}, \\ u=\varphi \quad \text{in }\partial \mathbb{R}_{+}^{n}, \\ \lim_{x_{n}\to +\infty }\frac{u(x)}{x_{n}}=c\geq 0, \end{gather*} has a positive solution $u$ satisfying $u(x)\geq cx_{n}+\rho _{0}(x)$ in $\mathbb{R}_{+}^{n}$, where $\rho _{0}$ is a fixed positive continuous bounded harmonic function in $\overline{\mathbb{R}_{+}^{n}}$. In the sublinear case where $f(x,u)=p(x)u^{\alpha}$, $0<\alpha \leq 1$, Lair and Wood \cite{l1} studied the existence of positive large solutions and bounded ones for the equation \eqref{E}. In particular they proved the existence of entire bounded nonnegative solutions in $\mathbb{R}^{n}$ provided that $p$ is locally h\"{o}lder continuous and satisfies $\int_{0}^{\infty }t\max_{|x|=t}(p(x))dt<\infty$. This result was extended by Bachar and Zeddini in \cite{b2} to more general function $f(x,u)=q(x)g(u)$. More precisely it is shown in \cite{b2} that the equation \eqref{E} has at least one positive continuous bounded solution in $\mathbb{R}^{n}$, provided that the Green potential of $q$ is continuous bounded in $\mathbb{R}^{n}$ and for all $\alpha >0$, there exists a constant $k>0$ such that the function $x\to kx-g(x)$ is nondecreasing on $[\alpha ,\infty )$. In this work, we will give two existence results for the problem \eqref{Pab}. For this aim, we fix a positive harmonic function $h_{0}$ in $D$, which is continuous and bounded in $\overline{D}$ such that $\lim_{|x|\to +\infty}h_{0}(x)=0$, whenever $n\geq 3$. We suppose that the function $f$ satisfies combinations of the following hypotheses: \begin{enumerate} \item[(H1)] $f:D\times (0,+\infty )\to [0,+\infty )$ is measurable, continuous with respect to the second variable. \item[(H2)] There exists a nonnegative measurable function $\theta$ on $D\times (0,+\infty )$ such that the function $t\mapsto \theta (x,t)$ is nonincreasing on $(0,+\infty )$, and satisfies \begin{equation*} f(x,t)\leq \theta (x,t),\text{ for \ }( x,t)\in D\times (0,+\infty ). \end{equation*} \item[(H3)] The function $\psi$ defined on $D$ by $\psi (x)=\frac{\theta (x,h_{0}(x))}{ h_{0}(x)}$ belongs to the class $K^{\infty }(D)$. \item[(H4)] For each $\alpha \geq 0$ and $\beta \geq 0$ with $\alpha +\beta >0$, there exists a nonnegative function $q_{\alpha,\beta }=q\in K^{\infty }(D)$ such that for each $x\in D$ and $t\geq s\geq \alpha h_{0}(x)+\beta h(x)$, we have \begin{gather} f(x,t)-f(x,s)\leq (t-s)q(x), \label{e1.1}\\ f(x,t)\leq tq(x). \end{gather} \end{enumerate} For the rest of this paper, we denote by $H_{D}\varphi$ the bounded continuous solution of the Dirichlet problem $$\label{e1.3} \begin{gathered} \Delta w=0\quad \text{in }D,\\ w\big|_{\partial D}=\varphi , \\ \lim_{|x|\to +\infty }\frac{w(x)}{h(x)}=0, \end{gathered}$$ where $\varphi$ is a nonnegative continuous function on $\partial D$ and $h$ is the harmonic function given by \eqref{e2.2}. The outline of this paper is as follows. In the second section we recall and improve some useful results concerning estimates on the Green function $G_{D}$ of the Laplace operator $\Delta$ in $D$ and some properties of functions belonging to the Kato class $K^{\infty }(D)$. In section 3, we will prove a first existence result for the problem \eqref{Pab}, by using the Schauder fixed-point theorem. More precisely, we prove the following \begin{theorem} \label{thm1} Under the assumptions (H1)--(H3), there exists a constant $c>1$ such that if $\varphi \geq ch_{0}$ on $\partial D$, then for each $\lambda \geq 0$, the problem \eqref{Pab} with $\alpha=\beta=1$ has a positive continuous solution $u$ satisfying for each $x\in D$, \begin{equation*} \lambda h(x)+h_{0}(x)\leq u(x)\leq \lambda h(x)+H_{D}\varphi (x). \end{equation*} \end{theorem} In the last section, we use a potential theory approach to prove a second existence result for the problem \eqref{Pab}. More precisely, we will prove the following result. \begin{theorem} \label{thm2} Under the assumptions (H1) and (H4), if $\alpha \geq 0$ and $\beta \geq 0$ with $\alpha +\beta >0$, then there exists a constant $c_{1}>1$ such that if $\varphi \geq c_{1}h_{0}$ on $\partial D$ and $\lambda \geq c_{1}$, the problem \eqref{Pab} has a positive continuous solution $u$ satisfying: For each $x\in D$, \begin{equation*} \alpha h_{0}(x)+\beta h(x)\leq u(x) \leq \alpha H_{D}\varphi (x)+\beta \lambda h(x). \end{equation*} \end{theorem} \subsection*{Notation and preliminaries} Throughout this paper, we will adopt the following notation. \begin{enumerate} \item[i.] $D$ is an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$) such that the complement of $\overline{D}$ in $\mathbb{R}^{n}$, $\overline{D}^{c}=\bigcup_{j=1}^{d}D_{j}$ where $D_{j}$ is a bounded $C^{1,1}$-domain and $\overline{D}_{i}\bigcap \overline{D}_{j}=\emptyset$, for $i\neq j$. \item[ii.] For a metric space $S$, we denote by $\mathcal{B}(S)$ the set of Borel measurable functions and $\mathcal{B}_{b}(S)$ the set of bounded ones. $\mathcal{C}(S)$ will denote the set of continuous functions on $S$. The exponent + means that only the nonnegative functions are considered. \item[iii.] $\mathcal{C}_{0}(\overline{D})=\{f\in \mathcal{C}(\overline{D}):\lim_{|x|\to +\infty }f(x)=0\}$. \item[iv.] $\mathcal{C}_{b}(D)=\{f\in \mathcal{C}(D):f \text{ is bounded in }D\}$. We note that $\mathcal{C}_{0}(\overline{D})$ and $\mathcal{C}_{b}(D)$ are two Banach spaces endowed with the uniform norm \begin{equation*} \|f\|_{\infty }=\sup_{x\in D}|f( x)|. \end{equation*} \item[v.] For $x\in D$, we denote by $\delta _{D}(x)$ the distance from $x$ to $\partial D$, $\rho _{D}(x)=\frac{\delta _{D}(x)}{\delta _{D}(x)+1},\quad \lambda _{D}(x)=\delta _{D}(x)(\delta _{D}(x)+1).$ \item[vi.] Let $f$ and $g$ be two positive functions on a set $S$. We denote $f\sim g$, if there exists a constant $c>0$ such that \begin{equation*} \frac{1}{c}g(x)\leq f(x)\leq cg(x)\quad \text{for all }x\in S. \end{equation*} \item[vii.] For $f\in \mathcal{B}^{+}(D)$, we denote by $Vf$ the Green potential of $f$ defined on $D$ by \begin{equation*} Vf(x)=\int_{D}G_{D}(x,y)f(y)dy. \end{equation*} Recall that if $f\in L_{\rm loc}^{1}(D)$ and $Vf\in L_{\rm loc}^{1}(D)$, then we have in the distributional sense (see \cite[p. 52]{c1}) $$\label{e1.4} \Delta (Vf)=-f\quad \text{in }D.$$ Furthermore, we recall that for $f\in \mathcal{B}^{+}(D)$, the potential $Vf$ is lower semi-continuous in $D$ and if $f=f_{1}+f_{2}$ with $f_{1},f_{2}\in \mathcal{B}^{+}(D)$ and $Vf\in \mathcal{C}^{+}(D)$, then $Vf_{i}\in \mathcal{C}^{+}(D)$ for $i\in \{1,2\}$. \item[viii.] Let $(X_{t},t>0)$ be the Brownian motion in $\mathbb{R}^{n}$ and $P^{x}$ be the probability measure on the Brownian continuous paths starting at $x$. For $q\in \mathcal{B}^{+}(D)$, we define the kernel $V_{q}$ by $$\label{e1.5} V_{q}f(x)=E^{x}\Big(\int_{0}^{\tau_{D}} e^{-\int_{0}^{t}q(X_{s})ds}f(X_{t})dt\Big),$$ where $E^{x}$ is the expectation on $P^{x}$ and $\tau _{D}=\inf \{t>0:X_{t}\notin D\}$. If $q\in \mathcal{B}^{+}(D)$ such that $Vq<\infty$, the kernel $V_{q}$ satisfies the resolvent equation (see \cite{c1,m1}) $$\label{e1.6} V=V_{q}+V_{q}(qV)=V_{q}+V(qV_{q}).$$ So for each $u\in \mathcal{B}(D)$ such that $V(q|u|)<\infty$, we have $$\label{e1.7} (I-V_{q}(q.))(I+V(q.))u=(I+V(q.))(I-V_{q}(q.)) u=u.$$ \item[ix.] We recall that a function $f:[0,\infty ) \to \mathbb{R}$ is called completely monotone if $(-1)^{n}f^{(n)}\geq 0$, for each $n\in \mathbb{N}$. Moreover, if $f$ is completely monotone on $[0,\infty )$ then by \cite[Theorem 12a]{w1} there exists a nonnegative measure $\mu$ on $[0,\infty )$ such that \begin{equation*} f(t)=\int_{0}^{\infty }\exp (-tx)d\mu (x). \end{equation*} So, using this fact and the Holder inequality we deduce that if $f$ is completely monotone from $[0,\infty )$ to $(0,\infty)$, then $\log f$ is a convex function. \item[x.] Let $f \in \mathcal{B}^{+}(D)$ be such that $Vf<\infty$. From \eqref{e1.5}, it is easy to see that for each $x\in D$, the function $F:\lambda \to V_{\lambda q}f(x)$ is completely monotone on $[0,\infty )$. \item[xi.] Let $a\in \mathbb{R}^{n}\backslash \overline{D}$ and $r>0$ such that $\overline{B(a,r)}\subset \mathbb{R}^{n}\backslash \overline{D}$. Then we have \begin{gather*} G_{D}(x,y)=r^{2-n}G_{\frac{D-a}{r}}(\frac{x-a}{r},\frac{y-a}{r}), \quad\mbox{for }x,y\in D, \\ \delta _{D}(x)=r\delta _{\frac{D-a}{r}}(\frac{x-a}{r}),\quad \text{for }x\in D, \end{gather*} So without loss of generality, we may suppose that $\overline{B(0,1)}\subset \mathbb{R}^{n}\diagdown\overline{D}$. Moreover, we denote by $D^{\ast }$ the open set \begin{equation*} D^{\ast }=\{x^{\ast }\in B(0,1):x\in D\cup \{\infty \} \} , \end{equation*} where $x^{\ast }=x/|x|^{2}$ is the Kelvin inversion from $D\cup \{\infty \}$ onto $D^{\ast }$ (see \cite{b1,m3}). Then for $x,y\in D$, \begin{equation*} G_{D}(x,y)=|x|^{2-n}|y|^{2-n}G_{D^{\ast }}(x^{\ast },y^{\ast }). \end{equation*} \end{enumerate} Also we mention that the letter $C$ will denote a generic positive constant which may vary from line to line. \section{Properties of the Green function and the Kato class} In this section, we recall and improve some results concerning the Green function $G_{D}(x,y)$ and the Kato class $K^{\infty }(D)$, which are stated in \cite{b1} for $n\geq 3$ and in \cite{m3} for $n=2$. \begin{theorem}[3G-Theorem] \label{3G-thm} There exists a constant $C_{0}>0$ depending only on $D$ such that for all $x,y$ and $z$ in $D$ \begin{equation*} \frac{G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}\leq C_{0}\Big(\frac{\rho _{D}(z)}{\rho _{D}(x)}G_{D}(x,z)+\frac{\rho _{D}(z)}{\rho _{D}(y)}G_{D}(y,z)\Big). \end{equation*} \end{theorem} \begin{proposition} \label{prop1} On $D^{2}$ (that is $x,y\in D$), we have \begin{equation*} G_{D}(x,y)\sim \begin{cases} \frac{1}{|x-y|^{n-2}}\min \Big(1,\frac{\lambda _{D}(x)\lambda _{D}(y)}{|x-y|^{2}}\Big),& n\geq 3, \\ \log(1+\frac{\lambda _{D}(x)\lambda _{D}(y)}{|x-y|^{2}}),& n=2. \end{cases} \end{equation*} Moreover, for $M>1$ and $r>0$ there exists a constant $C>0$ such that for each $x\in D$ and $y\in D$ satisfying $|x-y|\geq r$ and $|y|\leq M$, we have $$\label{e2.1} G_{D}(x,y)\leq C\frac{\rho _{D}(x)\rho _{D}(y)}{|x-y|^{n-2}}.$$ \end{proposition} \begin{definition} \label{def1} \rm A Borel measurable function $q$ in $D$ belongs to the Kato class $K^{\infty}(D)$ if $q$ satisfies \begin{gather*} \lim_{\alpha \to 0}\,(\sup_{x\in D}\int_{D\cap B(x,\alpha )}\frac{\rho _{D}(y)}{\rho _{D}(x)} G_{D}(x,y)|q(y)|dy)=0, \\ \lim_{M\to \infty }\,(\sup_{x\in D}\int_{D\cap (|y|\geq M)}\frac{\rho _{D}(y)}{\rho _{D}(x)}G_{D}(x,y)|q(y)|dy)=0. \end{gather*} \end{definition} In this paper, $h$ denotes the function defined, on $D$, by $$\label{e2.2} h(x)=c_{n}|x|^{2-n}G_{D^{\ast }}( x^{\ast },0)=c_{n}\lim_{|y|\to +\infty }|y|^{n-2}G_{D}(x,y),$$ where $c_{n}=\begin{cases} 2\pi & \text{for } n=2, \\ \frac{4\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}-1)} &\text{for } n\geq 3. \end{cases}$. Then we have the following statement. \begin{proposition} \label{prop2} The function $h$ defined by \eqref{e2.2} is harmonic in $D$ and satisfies $\lim_{x\to z\in \partial D}h(x)=0$, \begin{gather*} \lim_{|x|\to \infty }\frac{h(x)}{\log|x|}=1\quad \text{ for }n=2, \\ \lim_{|x|\to \infty }h(x)=1\quad \text{for }n\geq 3. \end{gather*} Moreover, $$\label{e2.3} h(x)\sim \begin{cases} \rho _{D}(x) &\text{for } n\geq 3, \\ \log(1+\rho _{D}(x))&\text{for }n=2. \end{cases}$$ \end{proposition} The proof of the above proposition can be found in \cite[Lemma 4.1]{m3} and in \cite{m5}. \begin{remark}[{\cite[p.427]{d1}}] \label{rmk1} \rm The function $H_{D}\varphi$ defined in \eqref{e1.3} belongs to $\mathcal{C}(\overline{D}\cup \{\infty \} )$ and satisfies \begin{equation*} \lim_{|x|\to +\infty }| x|^{n-2}H_{D}\varphi (x)=C>0. \end{equation*} \end{remark} In the sequel, we use the notation \begin{gather} \|q\|_{D}=\sup_{x\in D}\int_{D}\frac{ \rho _{D}(y)}{\rho _{D}(x)}G_{D}(x,y)|q(y)|dy \\ \alpha _{q}=\sup_{x,y\in D}\int_{D}\frac{ G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}|q(z)|dz. \label{e2.4} \end{gather} It is shown in \cite{b1,m3} that if $q\in K^{\infty }(D)$, then $$\label{e2.5} \|q\|_{D}<\infty.$$ Now, we remark that from the 3G-Theorem, \begin{equation*} \alpha _{q}\leq 2C_{0}\|q\|_{D}, \end{equation*} where $C_{0}$ is the constant. Next, we prove that $\alpha _{q}\sim \|q\|_{D}$. \begin{proposition} \label{prop3} The following assertions hold \begin{enumerate} \item[(i)] For any nonnegative superharmonic function $v$ in $D$ and any $q$ in $K^{\infty }(D)$, $$\label{e2.6} \int_{D}G_{D}(x,y)v(y)|q(y)|dy\leq \alpha _{q}v(x),\quad \forall x\in D.$$ \item[(ii)] There exists a constant $C>0$ such that for each $q\in K^{\infty }(D)$, \begin{equation*} C\|q\|_{D}\leq \alpha _{q}. \end{equation*} \end{enumerate} \end{proposition} \begin{proof} (i) Let $v$ be a nonnegative superharmonic function in $D$. Then by \cite[Theorem 2.1]{p1} there exists a sequence $(f_{k})_{k}$ of nonnegative measurable functions in $D$ such that the sequence $(v_{k})_{k}$ defined on $D$ by \begin{equation*} v_{k}(y):=\int_{D}G_{D}(y,z)f_{k}(z)dz \end{equation*} increases to $v$. Since for each $x\in D$, we have \begin{equation*} \int_{D}G_{D}(x,y)v_{k}(y)|q(y)|dy\leq \alpha _{q}v_{k}(x), \end{equation*} the result follows from the monotone convergence theorem. $(ii)$ We will discuss two cases: Case 1 ($n\geq 3$). Using Fatou's Lemma and \eqref{e2.2} we obtain $\int_{D}\frac{h(z)}{h(x)}G_{D}(x,z)|q(z)|dz \leq \liminf_{|y|\to +\infty} \int_{D}\frac{G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}|q(z)|dz\leq \alpha _{q}.$ Hence, the result follows from \eqref{e2.3}. \noindent Case 2 ($n=2$). Let $\varphi _{1}$ be a positive eigenfunction associated to the first eigenvalue of the Laplacian in $D^{\ast }$. From \cite[Proposition 2.6]{m2}, we have \begin{equation*} \varphi _{1}(\xi )\sim \delta _{D^{\ast }}(\xi ),\quad \forall \xi \in D^{\ast }. \end{equation*} Let $v(x)=\varphi _{1}(x^{\ast })$ for $x\in D$. Then $v$ is superharmonic in $D$ and \begin{equation*} v(x)\sim \delta _{D^{\ast }}(x^{\ast })\sim \rho_{D}(x). \end{equation*} Applying the assertion (i) to this function $v$ we deduce the result. \end{proof} \begin{proposition}[\cite{b1,m3}] \label{prop4} Let $q$ be a function in $K^{\infty }(D)$ and $v$ be a positive superharmonic function in $D$. \begin{enumerate} \item[(a)] Let $x_{0}\in \overline{D}$. Then \begin{gather} \lim_{r\to 0}(\sup_{x\in D}\int_{B(x_{0},r)\cap D}\frac{v(y)}{ v(x)}G_{D}(x,y)|q(y)|dy)=0, \label{e2.7} \\ \lim_{M\to +\infty }(\sup_{x\in D}\int_{(|y|\geq M)\cap D}\frac{ v(y)}{v(x)}G_{D}(x,y)|q(y)|dy)=0. \label{e2.8} \end{gather} \item[(b)] The potential $Vq$ is in $\mathcal{C}_{b}( D)$, $\lim_{x\to z\in \partial D}Vq(x)=0$, and for $n\geq 3$, $\lim_{|x| \to +\infty }Vq(x)=0$. \item[(c)] The function $x\to \frac{\delta_{D}(x)}{|x|^{n-1}}q(x)$ is in $L^{1}(D)$. \end{enumerate} \end{proposition} \begin{example} \label{ex1} \rm Let $p>n/2$ and $\gamma ,\mu \in \mathbb{R}$ such that $\gamma <2-\frac{n}{p}<\mu$. Then using the H\"{o}lder inequality and the same arguments as in \cite[Proposition 3.4]{b1} and \cite[Proposition 3.6]{m3}, it follows that for each $f\in L^{p}(D)$, the function defined in $D$ by $\frac{f(x)}{|x|^{\mu -\gamma }(\delta _{D}(x))^{\gamma }}$ belongs to $K^{\infty }(D)$. Moreover, by taking $p=+\infty$, we find again the results of \cite{b1,m3}. \end{example} \begin{proposition} \label{prop5} Let $v$ be a nonnegative superharmonic function in $D$ and $q\in K_{+}^{\infty }(D)$. Then for each $x\in D$ such that $00$. Let $x\in B(x_{0},r)\cap D$ and $p\in \Gamma _{q}$. Since $h_{0}$ is bounded, for $M>0$ we have \begin{align*} \frac{1}{\|h_{0}\|_{\infty}}|Lp(x)-Lp(x_{0})| &\leq \int_{D}|G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy\\ &\leq 2\sup_{z\in D}\int_{B(x_{0},2r)\cap D}G_{D}(z,y)q(y)dy \\ &\quad +2\sup_{z\in D}\int_{(|y|\geq M)\cap D}G_{D}(z,y)q(y)dy \\ &\quad +\int_{\Omega }|G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy, \end{align*} where $\Omega =B^{c}(x_{0},2r)\cap B(0,M)\cap D$. On the other hand, for every $y\in \Omega$ and $x\in B(x_{0},r)\cap D$, using \eqref{e2.1}, we obtain \begin{align*} |G_{D}(x,y)-G_{D}(x_{0},y)| &\leq C[\frac{\rho_{D}(x)}{|x-y|^{n-2}} +\frac{\rho_{D}(x_{0})}{|x_{0}-y|^{n-2}}]\rho_{D}(y)\\ &\leq C\delta _{D}(y) \leq C\frac{\delta _{D}(y)}{|y|^{n-1}}. \end{align*} Now, since $G_{D}$ is continuous outside the diagonal, we deduce by the dominated convergence theorem and Proposition \ref{prop4} (c) that \begin{equation*} \int_{\Omega }| G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy\to 0\quad \text{as }|x-x_{0}|\to 0. \end{equation*} So, using Proposition \ref{prop4}(a) for $v\equiv 1$, we deduce that $|Lp(x)-Lp(x_{0})|\to 0$ as $|x-x_{0}|\to 0$, uniformly for all $p\in \Gamma_{q}$. On the other hand, on $D$, we have $$\label{e3.1} |Lp(x)|\leq \|h_{0}\|_{\infty }Vq(x),$$ which tends to zero as $x\to \partial D$. Hence, $L(\Gamma _{q})$ is equicontinuous on $\overline{D}$. Next, we shall prove that $L(\Gamma _{q})$ is equicontinuous at $\infty$. First, we claim that \begin{equation*} \lim_{_{|x|\to \infty }}Lp(x) =\begin{cases} 0 &\text{for }n\geq 3, \\ \int_{D}h_{0}(y)p(y)h(y)dy &\text{for }n=2. \end{cases} \end{equation*} Using \eqref{e3.1} and Proposition \ref{prop4}(b), we obtain $Lp(x)\to 0$ as $|x|\to \infty$, for $n\geq 3$, uniformly in $p\in \Gamma _{q}$. Finally, we assume that $n=2$ and we put $l=\int_{D}h_0(y)p(y)h(y)dy$. Since $\lim_{|x|\to +\infty}G_{D}(x,y)=h(y)$, then using Fatou's lemma and Proposition \ref{prop4}(b), we obtain \begin{align*} |l|&\leq \int_{D}h_0(y) q(y)h(y)dy\\ &\leq \liminf_{|x|\to+\infty }\int_{D}G(x,y)h_0(y)q(y)dy\\ &\leq \|h_0\|_{\infty }\|Vq\|_{\infty }<+\infty . \end{align*} Now, we shall prove that $\lim_{|x|\to +\infty}Lp(x)=l$. Let $\varepsilon >0$, then by \eqref{e2.8}, there exists $M>1$ such that for each $x\in D$ with $|x|\geq 1+M$ we have \begin{align*} |Lp(x)-l|&\leq \int_{D}|G_{D}(x,y)-h(y)| h_{0}(y)q(y)dy \\ &\leq \varepsilon +\int_{B(0,M)\cap D}|G_{D}(x,y)-h(y)|h_{0}(y) q(y)dy. \end{align*} On the other hand, using \eqref{e2.1}, for $y\in B(0,M)\cap D$ and $|x|\geq 1+M$, we have $|G_{D}(x,y)-h(y)|h_0(y)\leq C(\frac{ \delta _{D}(y)}{|y|}+h(y)).$ We deduce from Proposition \ref{prop4}(c) and Lebesgue's theorem that $\lim_{|x|\to +\infty}Lp(x)=l$, uniformly in $p\in \Gamma _{q}$. Thus by Ascoli's theorem $F_{q}$ is relatively compact in $\mathcal{C}(\overline{D}\cup \{\infty \} )$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] We shall use a fixed-point argument. Let $c=1+\alpha _{\psi }$, where $\alpha _{\psi }$ is the constant defined by \eqref{e2.4} associated to the function $\psi$ given in (H3) and suppose that \begin{equation*} \varphi (x)\geq ch_{0}(x),\quad \forall x\in \partial D. \end{equation*} Since $h_{0}$ is a harmonic function in $D$, continuous and bounded in $\overline{D}$, then the function $w:=H_{D}\varphi -ch_{0}$ is a solution to the problem \begin{gather*} \Delta w=0\quad \text{in }D, \\ w\big|_{\partial D}=\varphi -ch_{0}\geq 0, \\ \lim_{|x|\to +\infty }\frac{w(x)}{h(x)}=0, \end{gather*} and by the maximum principle it follows that $$\label{e3.2} H_{D}\varphi (x)\geq ch_{0}(x),\quad \forall x\in \overline{D}.$$ Let $\lambda \geq 0$ and let $\Lambda$ be the non-empty closed bounded convex set \begin{equation*} \Lambda =\{v\in C(\overline{D}\cup \{\infty \} ):h_{0}\leq v\leq H_{D}\varphi \} . \end{equation*} Let $S$ be the operator defined on $\Lambda$ by \begin{equation*} Sv(x)=H_{D}\varphi (x)-\int_{D}G_{D}(x,y)f(y,v(y)+\lambda h(y))dy. \end{equation*} We shall prove that the family $S\Lambda$ is relatively compact in $C(\overline{D}\cup \{\infty \} )$. Let $v\in \Lambda$, then by (H2) and (H3) and the fact that $h_{0}$ is positive in $D$, we have for each $y\in D$, $\frac{1}{h_{0}(y)}f(y,v(y)+\lambda h(y))\leq \frac{\theta (y,h_{0}(y))}{ h_{0}(y)}=\psi (y).$ Hence, we deduce that the function \begin{equation*} y\mapsto \frac{1}{h_{0}(y)}f(y,v(y) +\lambda h(y))\in \Gamma _{\psi }. \end{equation*} It follows that the family \begin{equation*} \{\int_{D}G_{D}(.,y)f(y,v(y) +\lambda h(y))dy:v\in \Lambda \} \subseteq F_{\psi }. \end{equation*} Thus, from Lemma \ref{lem1}, the family $\{\int_{D}G_{D}(.,y)f(y,v(y)+\lambda h(y))dy:v\in \Lambda \}$ is relatively compact in $C(\overline{D}\cup \{\infty \} )$. Since $H_{D}\varphi$\ $\in C(\overline{D}\cup \{\infty \} )$, we deduce that the family $S(\Lambda)$ is relatively compact in $C(\overline{D}\cup \{\infty\} )$. Next, we shall prove that $S$ maps $\Lambda$ to itself. It's clear that for all $v\in \Lambda$ we have $Sv(x) \leq H_{D}\varphi (x),\forall x\in D$. Moreover, from hypothesis (H2) and \eqref{e2.6}, it follows that \begin{align*} \int_{D}G_{D}(x,y)f(y,v(y)+\lambda h(y))dy &\leq \int_{D}G_{D}(x,y)\theta (y,h_{0}(y))dy\\ &=\int_{D}G_{D}(x,y)\psi (y)h_{0}(y)dy \\ & \leq \alpha _{\psi}h_{0}(x). \end{align*} Hence, using \eqref{e3.2} we obtain $Sv(x)\geq H_{D}\varphi (x)-\alpha _{\psi }h_{0}(x)\geq h_{0}(x)$, which proves that $S(\Lambda )\subset \Lambda$. Now, we prove the continuity of the operator $S$ in $\Lambda$ in the supremum norm. Let $(v_{k})_{k}$ be a sequence in $\Lambda$ which converges uniformly to a function $v$ in $\Lambda$. Then, for each $x\in D$, we have $|Sv_{k}(x)-Sv(x)|\leq \int_{D}G_{D}(x,y)|f( y,v_{k}(y)+\lambda h(y))-f(y,v(y)+\lambda h(y))|dy.$ On the other hand, by hypothesis (H2), we have $|f(y,v_{k}(y)+\lambda h(y)) -f(y,v(y)+\lambda h(y))|\leq 2h_{0}(y)\psi (y)\leq 2\|h_{0}\|_{\infty }\psi (y).$ Since by Proposition \ref{prop4}(b), $V\psi$ is bounded, we conclude by the continuity of $f$ with respect to the second variable and by the dominated convergence theorem that for all $x\in D$, \begin{equation*} Sv_{k}(x)\to Sv(x) \quad \mbox{as }k\to +\infty . \end{equation*} Consequently, as $S(\Lambda )$ is relatively compact in $C(\overline{D}\cup \{\infty \} )$, we deduce that the pointwise convergence implies the uniform convergence, namely, \begin{equation*} \|Sv_{k}-Sv\|_{\infty }\to 0\quad\mbox{as } k\to +\infty . \end{equation*} Therefore, $S$ is a continuous mapping of $\Lambda$ to itself. So since $S\Lambda$ is relatively compact in $C(\overline{D}\cup \{\infty \} )$ it follows that $S$ is compact mapping on $\Lambda$. Finally, the Schauder fixed-point theorem implies the existence of $v\in \Lambda$ such that \begin{equation*} v(x)=H_{D}\varphi (x)-\int_{D}G_{D}( x,y)f(y,v(y)+\lambda h(y))dy. \end{equation*} Put $u(x)=v(x)+\lambda h(x)$, for $x\in D$. Then $u\in C(\overline{D})$ and $u$ satisfies $$\label{e3.3} u=H_{D}\varphi +\lambda h-\int_{D}G_{D}(.,y)f( y,u(y))dy.$$ Now, we verify that $u$ is a solution of \eqref{Pab} with $\alpha=\beta=1$. Since $\psi \in K^{\infty }(D)$, it follows from Proposition \ref{prop4}(c), that $\psi \in L_{\rm loc}^{1}(D)$. Furthermore, by hypotheses (H2) and (H3) we have $f(.,u)\leq h_{0}\psi$. This shows that $f(.,u)\in L_{\rm loc}^{1}(D)$ and $V(f(.,u))\in F_{\psi }$. Then, from Lemma \ref{lem1}, we have $V(f(.,u))\in C(\overline{D}\cup \{\infty \} )\subset L_{\rm loc}^{1}(D)$. Thus, by applying $\Delta$ on both sides of \eqref{e3.3} and using \eqref{e1.4}, we obtain that $u$ satisfies the elliptic equation (in the sense of distributions) \begin{equation*} \Delta u=f(.,u)\quad \text{in }D\,. \end{equation*} Since $H_{D}\varphi =\varphi$ on $\partial D,\lim_{x\to z\in \partial D}h(x)=0$, and $\lim_{x\to z\in \partial D}V(f(.,u))(x)=0$, we conclude that $\lim_{x\to z\in \partial D}u(x)=\varphi (z)$. On the other hand, since \begin{equation*} \lambda h(x)+h_{0}(x)\leq u(x)\leq \lambda h(x)+H_{D}\varphi (x) \end{equation*} and $\lim_{|x|\to +\infty }\frac{ H_{D}\varphi (x)}{h(x)}=\lim_{|x| \to +\infty }\frac{h_{0}(x)}{h(x)}=0$, we deduce $\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\lambda$. This completes the proof. \end{proof} \begin{example} \label{ex2} \rm Let $D=B^{c}(0,1)$, $p>\frac{n}{2}$, $\sigma >0$ and $\nu >0$. Let $\varphi$ and $g$ in $\mathcal{C}^{+}(\partial D)$ and put $h_{0}=H_{D}g$. Then from \cite[p. 258]{a1}, there exists a constant $c_{0}>0$ such that for each $x\in D$, \begin{equation*} \frac{c_{0}(|x|-1)}{|x|^{n-1}}\leq h_{0}(x) \end{equation*} Moreover, suppose that the function $f$ satisfies (H1) and \begin{equation*} f(x,t)\leq t^{-\sigma }\frac{v(x)}{| x|^{\nu -1+n(\sigma +1)}(|x|-1) ^{1-2\sigma -\frac{n}{p}}}, \end{equation*} where $v\in L_{+}^{p}(D)$. Then, there exists a constant $c>1$ such that if $\varphi \geq cg$ on $\partial D$, the problem \eqref{Pab} with $\alpha=\beta=1$ has a positive solution $u$ in $\mathcal{C}(\overline{D})$ satisfying that for each $x\in D$, \begin{equation*} \lambda h(x)+h_{0}(x)\leq u(x)\leq \lambda h(x)+H_{D}\varphi (x), \end{equation*} where $h$ is the function given by \eqref{e2.2}. Indeed, (H1) and (H2) are satisfied and by taking $\gamma =2-\sigma -\frac{n}{p}$and $\mu =2-\frac{n}{p}+\nu$ in Example \ref{ex1}, we deduce that the function $x\mapsto (h_{0}(x))^{-1-\sigma } \frac{v(x)}{|x|^{\nu -1+n(\sigma +1) }(|x|-1)^{1-2\sigma -\frac{n}{p}}} \in K^{\infty }(D),$ which implies that hypothesis (H3) is satisfied. \end{example} \section{Proof of Theorem \ref{thm2}} Recall that for a fixed nonnegative function $q\in K^{\infty}(D)$, we have defined the set $\Gamma _{q}=\{p\in K^{\infty }(D):|p|\leq q\}$. Using Propositions \ref{prop3} and \ref{prop4}, with similar arguments as in \cite[Lemma 4.3]{m3}, we establish the following lemma. \begin{lemma} \label{lem2} Let $q$ be a nonnegative function in $K^{\infty }(D)$ and let $h$ be the function given by \eqref{e2.2}. Then the family of functions \begin{equation*} \mathfrak{F}_{q}(h)=\big\{\frac{1}{h}\int_{D}G(.,y)h(y)p(y)dy:p \in \Gamma _{q}\big\} \end{equation*} is uniformly bounded and equicontinuous in $\overline{D}\cup \{\infty\}$. Consequently, it is relatively compact in $\mathcal{C}_{0}(\overline{D})$. \end{lemma} \begin{proof}[Proof of Theorem \ref{thm2}] Let $\alpha \geq 0$, $\beta \geq 0$ with $\alpha +\beta >0$ and let $q:=q_{\alpha ,\beta }$ be the function in $K^{\infty }(D)$ given by (H4). Let $c_{1}:=e^{\alpha _{q}}>1$, where $\alpha _{q}$ is the constant given by \eqref{e2.4}. Suppose that \begin{equation*} \varphi (x)\geq c_{1}h_{0}(x),\quad \forall x\in \partial D. \end{equation*} Then by the maximum principle it follows that $$\label{e4.1} H_{D}\varphi (x)\geq c_{1}h_{0}(x),\quad \forall x\in \overline{D}.$$ Now, let $\lambda \geq c_{1}$ and put $$\label{e4.2} \begin{gathered} w(x):=\beta \lambda h(x)+\alpha H_{D}\varphi (x)\text{, for }x\in D, \\ v(x):=\alpha h_0 +\beta h(x)\text{, for }x\in D. \end{gathered}$$ Consider the nonempty convex set \begin{equation*} \Omega :=\{u\in \mathcal{B}(D):v\leq u\leq w\} . \end{equation*} Let $T$ be the operator defined on $\Omega$ by \begin{equation*} Tu(x):=w(x)-V_{q}(qw)(x) +V_{q}(qu-f(.,u))(x). \end{equation*} From hypothesis (H4) we have for each $u\in\Omega$ $$\label{e4.3} 0\leq f(.,u)\leq uq.$$ Let us prove that the operator $T$ maps $\Omega$ to itself. By \eqref{e2.6}, it follows that $$\label{e4.4} \int_{D}G_{D}(x,y)w(y)q(y)dy\leq \alpha _{q}w(x).$$ Since $w$ is a harmonic function in $D$ and $Vq<\infty$, by \eqref{e4.3} and Proposition \ref{prop5}, we have for each $x\in D$, $Tu(x)\geq w(x)-V_{q}(qw)(x) \geq e^{-\alpha _{q}}w(x)=e^{-\alpha _{q}}(\beta \lambda h(x)+\alpha H_{D}\varphi (x)).$ Therefore, as $\lambda \geq c_{1}$ and by \eqref{e4.1} we obtain $Tu(x)\geq \beta h(x)+\alpha h_{0}(x)=v(x).$ On the other hand, we have for each $x\in D$, $Tu(x)\leq w(x)-V_{q}(qw)(x)+V_{q}(qu)(x) \leq w(x).$ So $T(\Omega )\subset \Omega$. Now, let $u_{1},u_{2}\in \Omega$ such that $u_{1}\geq u_{2}$, then by (H4) we have $Tu_{1}-Tu_{2}=V_{q}(q[u_{1}-u_{2}]-[f(.,u_{1})-f(.,u_{2})])\geq 0.$ Hence, $T$ is a nondecreasing operator on $\Omega$. Next, we consider the sequence $(u_{m})_{m\in \mathbb{N}}$ defined by \begin{equation*} u_{0}=\beta h+\alpha h_{0}\quad\text{and}\quad u_{m+1}=Tu_{m}\quad \text{for } m\in \mathbb{N}. \end{equation*} Since $\Omega$ is invariant under $T$, we obtain $v=u_{0}\leq u_{1}\leq w$. Therefore, from the monotonicity of $T$ on $\Omega$, we have \begin{equation*} v=u_{0}\leq u_{1}\leq \dots \leq u_{m}\leq u_{m+1}\leq w. \end{equation*} Thus, from the monotone convergence theorem and the fact that $f$ is continuous with respect to the second variable, the sequence $(u_{m})_{m\in \mathbb{N}}$ converges to a function $u$ satisfying $$\label{e4.5} u=(I-V_{q}(q.))w+V_{q}(qu-f(.,u)).$$ By \eqref{e2.5} and \eqref{e2.6}, we obtain for each $x\in D$, \begin{equation*} 0\leq V(qu)(x)\leq V(qw)( x)\leq \alpha _{q}w(x)<\infty . \end{equation*} Applying $(I+V(q.))$ on both sides of \eqref{e4.5}, it follows from \eqref{e1.6} and \eqref{e1.7} that $$\label{e4.6} u=\beta \lambda h+\alpha H_{D}\varphi -V(f(.,u)).$$ Now, let us verify that $u$ is a solution of the problem \eqref{Pab}. Since $q\in K^{\infty }(D)$ then by Proposition \ref{prop4}, we obtain $q\in L_{\rm loc}^{1}(D)$. By \eqref{e4.3} we have $$\label{e4.7} f(.,u)\leq qu\leq qw.$$ Therefore, since $w$ is continuous in $D$, we obtain that $f(.,u)\in L_{\rm loc}^{1}(D)$. Using Proposition \ref{prop3} and \eqref{e4.7}, for each $x\in D$, we have $V(f(.,u))(x)\leq \int_{D}G_{D}(x,y)w(y)q(y)dy\leq \alpha _{q}w(x).$ Then $V(f(.,u))\in L_{\rm loc}^{1}(D)$. Thus, by applying $\Delta$ on both sides of \eqref{e4.6}, we deduce that $u$ is a solution of \begin{equation*} \Delta u=f(.,u)\quad \text{ in }D \end{equation*} (in the sense of distributions). Using \eqref{e4.7} we obtain that $f(.,u)\leq \beta \lambda hq+\alpha qH_{D}\varphi \\ \leq \beta \lambda hq+\alpha \|\varphi \|_{\infty }q\,.$ Let $g:=\beta \lambda hq+\alpha \|\varphi \| _{\infty }q$. Since $f(.,u)$ and $(g-f(.,u))$ are in $\mathcal{B}^{+}(D)$ then $V(f(.,u))$ and $V(g-f(.,u))$ are two lower semi-continuous functions. On the other hand, by Proposition \ref{prop4}(b) we have $V(q)\in \mathcal{C}(D)$ and by Lemma \ref{lem2} the function $\frac{1}{h}V(hq)\in \mathcal{C}_{0}(\overline{D})$. So $Vg$ is a continuous function. This implies that $V(g-f(.,u))=Vg-V(f(.,u))$ is also an upper semi-continuous function. Consequently $V(g-f(.,u))$ is in $\mathcal{C}(D)$. Thus $V(f(.,u))=Vg-V(g-f(.,u))\in \mathcal{C}(D)$. Therefore $u$ is in $\mathcal{C}(D)$. Now using Proposition \ref{prop3}(i) and the fact that $\lim_{x \to z\in \partial D}h(x)=0$ we deduce that $\lim_{x \to \partial D}V(hq)(x)=0$. In addition from Proposition \ref{prop4}(b) we have $\lim_{x\to \partial D}V(q)(x)=0$. So that $\lim_{x\to \partial D}V(g)(x)=0$. This in turn implies that $\lim_{x\to \partial D}V(f(.,u))=0$. Then by \eqref{e4.6}, we obtain that $u\big|_{\partial D}=\alpha \varphi$. On the other hand, we have \begin{equation*} \frac{1}{h}V(f(.,u))\leq \beta \lambda \frac{1}{h }V(hq)+\alpha \|\varphi \|_{\infty }\frac{1 }{h}Vq. \end{equation*} Using Propositions \ref{prop2} and \ref{prop4}(b), we obtain that $\frac{1}{h(x)}V(f(.,u))(x)$ tends to $0$ as $|x|\to +\infty$ and consequently $\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda$. Hence, $u$ is a positive continuous solution in $D$ of the problem \eqref{Pab}. This completes the proof. \end{proof} \begin{example} \label{ex3} \rm Let $D=B^{c}(0,1)$ and $0<\gamma \leq 1$. Let $p$ be a nonnegative function such that the function $q(x)=(\frac{|x|^{n-1}}{|x|-1})^{1-\gamma }p(x)$ is in $K^{\infty}(D)$. Let $\varphi \in \mathcal{C}^{+}(\partial D)$ and $h_{0}$ be a positive harmonic function in $D$, which belongs to $\mathcal{C}_{b}(\overline{D})$. Then, for each $\alpha \geq 0$ and $\beta \geq 0$ with $\alpha +\beta >0$, there exists a constant $c_{1}>1$ such that if $\varphi \geq c_{1}h_{0}$ on $\partial D$ and $\lambda \geq c_{1}$, the problem \begin{gather*} \Delta u=p(x)u^{\gamma }\quad \text{in }D, \\ u\big|_{\partial D}=\alpha \varphi ,\\ \lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda \geq 0, \end{gather*} has a positive continuous solution on $D$ satisfying that for each $x\in D$, \begin{equation*} \beta h(x)+\alpha h_{0}(x)\leq u(x)\leq \beta \lambda h(x)+\alpha H_{D}\varphi (x). \end{equation*} \end{example} \subsection*{Acknowledgements} The authors want to thank Professor Habib Maagli for his valuable discussions, also to the anonymous referee for his/her valuable suggestions and comments. \begin{thebibliography}{00} \bibitem{a1} D. Armitage and S. Gardiner, \emph{Classical Potential Theory}, Springer- Verlag 2001. \bibitem{a2} S. 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